Intrinsic and Tunable Superconducting Diode Effect in Quantum Spin Hall Systems

Abstract

Nonreciprocal dissipationless transport has long been sought for applications in superconducting technologies. Recently, it has been implemented by the so called superconducting diode effect. Such effect arises from an imbalance in critical supercurrents flowing in opposite directions. In this work, we theoretically demonstrate how the superconducting diode effect emerges in the quantum spin Hall phase when brought into full proximity with a superconductor. We explore two regimes: large and narrow quantum wells. In the former geometry, we show that the superconducting diode effect can be externally controlled using both magnetic and electric fields, achieving unit efficiency. In the latter regime, where tunneling between opposite edges may occur, we propose a mechanism for an intrinsic superconducting diode effect driven by edge reconstruction, which does not require external magnetic fields.

Figures (5)

• Figure 1: (a) Schematic of the system analyzed. The superconductor (SC) placed on top of the heterostructure representing the 2DTI induces a superconducting coupling to the two edges. The black arrows represent the spin of the edge states. The colored arrows indicate the direction of motion of spin-polarized edge channels. (b) Schematics of spin-flip scattering processes and edge reconstruction. In the sketch we show an example of edge reconstruction with two spin-flip tunneling events (we have sketched the situation for a a fixed $k$ and with $\theta_k=\chi_k$). The tunneling amplitude related to $\tau_f(1+b)c_{k\uparrow}^\dagger d_{k\downarrow} +h.c.$ is favored over the amplitude related to $\tau_f(1-b)c_{k\downarrow}^\dagger d_{k\uparrow} + h.c.$, which implies $b>0$. Here we assumed symmetric spin preserving processes, with $a=0$.
• Figure 2: SDE performance with inhomogeneous spin--orbit coupling. The panel (a) shows the rectification coefficient as a function of magnetic field $U_z/\Delta$ and Rashba inhomogeneity $\Delta/v_Fk_{0,d}$. A near unit efficiency can be achieved for $U_z \lesssim \Delta$. Here, $\Delta/v_Fk_{0,u} = 0.05$ and $\mu/\Delta = 6$. The other three panels show the bands with different current bias at the parameters corresponding to the white star ($U_z/\Delta=0.93$, $\Delta/v_Fk_{0,d}=0.17$) in panel (a). Panel (b) shows the unbiased bands illustrating how symmetry breaking affects the bands. Since the superconducting gap corresponding to $-\mu/\Delta$ is very close to zero, it can be closed by a small $q>0$. This is shown in panel (c). Injecting the same amount of current in the opposite direction (panel (d)) does not close any gap, and this lead to a rectification of the supercurrent with $\eta \simeq 1$.
• Figure 3: SDE performance with inhomogeneous chemical potentials. Panel (a) shows the rectification coefficient as a function of magnetic field $U_z/\Delta$ and spin--orbit coupling $\Delta/v_Fk_{0}$. A near unit efficiency can be achieved for $U_z \lesssim \Delta$ and $\Delta/v_Fk_{0} \simeq 0.25$. Here, $\delta\mu/\Delta=2, \mu/\Delta = 4$. The remaining three panels display the band structures under different current bias for the parameters indicated by the white star ($U_z/\Delta=0.93$, $\Delta/v_F k_{0}=0.25$) in panel (a). Panel (b) shows the unbiased bands and highlights the effect of symmetry breaking. Because the superconducting gap near $-\mu/\Delta$ is almost zero, a small positive momentum $q>0$ is sufficient to close it, as shown in panel (c). Injecting the same magnitude of current in the opposite direction (panel (d)) does not close any gap, leading to near‑perfect rectification of the supercurrent, $\eta \simeq 1$.
• Figure 4: SDE performance in the narrow well regime without edge reconstruction ($a=b=0$). Panel (a) shows the rectification coefficient as a function of spin--orbit strength $\Delta/v_Fk_0$ and chemical potential $\mu/\Delta$. Here, $U_z/\Delta = 0.85, \tau_f/\Delta = 0.6, \tau_p/\Delta=0.7$. The remaining three panels illustrate the band structures under different current biases for the parameters marked by the white star ($\mu/\Delta = 3.85$, $\Delta/v_F k_{0} = 0.25$). Panel (b) displays the unbiased bands, highlighting the impact of symmetry breaking (the difference between the values of the two gaps at $q=0$ in this case is smaller but is not zero). A small positive momentum $q>0$ suffices to fully close the superconducting gap near $-\mu/\Delta$, as shown in panel (c). Applying the same current in the opposite direction (panel (d)) leaves the gap open, resulting in rectification of the supercurrent.
• Figure 5: Intrinsic SDE in the narrow well regime with edge reconstruction. Panel (a) presents the rectification coefficient as a function of spin--orbit strength $\Delta/v_Fk_0$, and chemical potential $\mu/\Delta$. Here, we have set $b=0.2, a=0$$\tau_f/\Delta = 0.4, \tau_p/\Delta = 0.7$. The remaining three panels show the band structures under different values of current bias for the parameters indicated by the white star ($\mu/\Delta = 4.30$, $\Delta/v_F k_{0} = 0.25$) in panel (a). Panel (b) depicts the unbiased bands, emphasizing the role of symmetry breaking. A positive momentum $q>0$ is sufficient to close the superconducting gap near $-\mu/\Delta$, as illustrated in panel (c). Driving the same current in the opposite direction (panel (d)) leaves the gap open, thereby producing rectification of the supercurrent. In panel (a) we observe zero rectification at specific values of the couple $(\mu,1/v_Fk_0)$ despite the breaking of the symmetries. The dashed lines represent our theoretical explanation of why along these lines the rectification is zero despite the breaking of the two necessary symmetries (see supplementary information).